Anderson localization
This phenomenon is named after the American physicist P. W. Anderson, who was the first to suggest that electron localization is possible in a lattice potential, provided that the degree of randomness (disorder) in the lattice is sufficiently large, as can be realized for example in a semiconductor with impurities or defects.
In the original Anderson tight-binding model, the evolution of the wave function ψ on the d-dimensional lattice Zd is given by the Schrödinger equation where the Hamiltonian H is given by[2] where
In the strong scattering limit, the severe interferences can completely halt the waves inside the disordered medium.
For non-interacting electrons, a highly successful approach was put forward in 1979 by Abrahams et al.[3] This scaling hypothesis of localization suggests that a disorder-induced metal-insulator transition (MIT) exists for non-interacting electrons in three dimensions (3D) at zero magnetic field and in the absence of spin-orbit coupling.
Much further work has subsequently supported these scaling arguments both analytically and numerically (Brandes et al., 2003; see Further Reading).
[4] However, since 2 is the lower critical dimension of the localization problem, the 2D case is in a sense close to 3D: states are only marginally localized for weak disorder and a small spin-orbit coupling can lead to the existence of extended states and thus an MIT.
Most numerical approaches to the localization problem use the standard tight-binding Anderson Hamiltonian with onsite-potential disorder.
Characteristics of the electronic eigenstates are then investigated by studies of participation numbers obtained by exact diagonalization, multifractal properties, level statistics and many others.
Direct numerical solution of Maxwell equations to demonstrate Anderson localization of light has been implemented (Conti and Fratalocchi, 2008).
It has also been observed by localization of a Bose–Einstein condensate in a 1D disordered optical potential (Billy et al., 2008; Roati et al., 2008).
Anderson localization of elastic waves in a 3D disordered medium has been reported (Hu et al., 2008).
The existence of Anderson localization for light in 3D was debated for years (Skipetrov et al., 2016) and remains unresolved today.
Reports of Anderson localization of light in 3D random media were complicated by the competing/masking effects of absorption (Wiersma et al., 1997; Storzer et al., 2006; Scheffold et al., 1999; see Further Reading) and/or fluorescence (Sperling et al., 2016).
Recent experiments (Naraghi et al., 2016; Cobus et al., 2023) support theoretical predictions that the vector nature of light prohibits the transition to Anderson localization (John, 1992; Skipetrov et al., 2019).
This approximation is repaired in maximal entropy random walk, also repairing the disagreement: it turns out to lead to exactly the quantum ground state stationary probability distribution with its strong localization properties.
